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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Convergence rates and source conditions for Tikhonov regularization with sparsity constraints

D. A. Lorenz1

1 Zentrum für Technomathematik, Fachbereich 3, Universität Bremen, PO Box 330440, 28334 Bremen, Germany. Email:

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 16, Issue 5, Pages 463–478, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/JIIP.2008.025, September 2008

Publication History:
Received:
2008-01-25
Revised:
2008-04-23
Published Online:
2008-09-12

Abstract

This paper addresses the regularization by sparsity constraints by means of weighted ℓp penalties for 0 ≤ p ≤ 2. For 1 ≤ p ≤ 2 special attention is payed to convergence rates in norm and to source conditions. As main results it is proven that one gets a convergence rate of in the 2-norm for 1 < p ≤ 2 and in the 1-norm for p = 1 as soon as the unknown solution is sparse. The case p = 1 needs a special technique where not only Bregman distances but also a so-called Bregman-Taylor distance has to be employed.

For p < 1 only preliminary results are shown. These results indicate that, different from p ≥ 1, the regularizing properties depend on the interplay of the operator and the basis of sparsity. A counterexample for p = 0 shows that regularization need not to happen.

Key words.: Sparsity constraint; ill-posed problems; Tikhonov regularization

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